You Have Decided to Use the Dickey Fuller (DF)test on the United

You have decided to use the Dickey Fuller (DF)test on the United States aggregate
unemployment rate (sample period 1962:I - 1995:IV).As a result, you estimate the
following AR(1)model $\begin{aligned}{\widehat{\Delta \text { UrateUS }_{t}}}{=} & 0.114-0.024 \text { UrateUS }_{\mathrm{t}-1}, R^{2}=0.0118, \text { SER }=0.3417 \\& (0.121)(0.019)\end{aligned}$
You recall that your textbook mentioned that this form of the AR(1)is convenient
because it allows for you to test for the presence of a unit root by using the t- statistic of
the slope.Being adventurous, you decide to estimate the original form of the AR(1)
instead, which results in the following output $\begin{aligned}{\widehat{\Delta \text { UrateUS }_{t}}}{=} & 0.114-0.024 \text { UrateUS }_{\mathrm{t}-1}, R^{2}=0.0118, \text { SER }=0.3417 \\& (0.121)(0.019)\end{aligned}$

You are surprised to find the constant, the standard errors of the two coefficients, and the SER unchanged, while the regression $\mathrm { R } ^ { 2 }$ increased substantially. Explain this increase in the regression $\mathrm { R } ^ { 2 }$ . Why should you have been able to predict the change in the slope coefficient and the constancy of the standard errors of the two coefficients and the SER?